Problem 73 - Master Resource Book in JEE Main Physics

Question

Under a pressure head, the rate of orderly volume flow of liquid through a capillary tube is \(Q\). If the length of capillary tube were doubled and the diameter of the bore is halved, the rate of flow would become (a) \(\frac{Q}{4}\) (b) \(16 Q\) (c) \(\frac{Q}{8}\) (d) \(\frac{0}{32}\)

Step-by-Step Solution

Answer

The rate of flow becomes \( \frac{Q}{32} \).

Step 1: Understand the Original Equation

The rate of volume flow of a liquid through a capillary tube is given by Poiseuille's Law, which is \( Q = \frac{\pi r^4 \Delta P}{8 \eta L} \), where \( r \) is the radius, \( \Delta P \) is the pressure difference, \( \eta \) is the viscosity, and \( L \) is the length of the tube.

Step 2: Applying Changes to the Length

If the length \( L \) of the capillary tube is doubled, then \( L \) becomes \( 2L \). According to Poiseuille’s Law, the flow rate \( Q \) is inversely proportional to length. Thus, the new flow rate considering length change only is \( Q' = \frac{Q}{2} \).

Step 3: Applying Changes to the Radius

If the diameter is halved, the radius \( r \) also halves (\( r \rightarrow \frac{r}{2} \)). Since the flow rate is directly proportional to \( r^4 \), the new flow rate when the radius is halved becomes \( Q'' = Q \left( \frac{1}{2} \right)^4 = \frac{Q}{16} \).

Step 4: Combine the Effects of Both Changes

Now, consider both changes - the new flow rate \( Q''' \) considering both doubled length and halved radius is \( Q''' = \left( \frac{Q}{16} \right) / 2 = \frac{Q}{32} \). This calculation uses the new flow rate from Step 3 divided by 2 due to the doubled length.

Key Concepts

Capillary Flow Rate

Capillary flow rate is crucial for understanding how liquid moves through narrow tubes. According to Poiseuille’s Law, the flow rate, represented by \( Q \), depends on several key factors: the radius of the capillary tube \( r \), the pressure difference \( \Delta P \) across the tube, the viscosity \( \eta \) of the fluid, and the length \( L \) of the tube.

• The radius is particularly influential because the flow rate varies with the fourth power of the radius, \( r^4 \). This means even a small change in radius has a huge impact on flow rate.
• The relationship is mathematically expressed as \( Q = \frac{\pi r^4 \Delta P}{8 \eta L} \). This formula shows us that wider tubes allow much more liquid to pass.
• As the radius decreases, the flow rate decreases dramatically, and vice versa.

Understanding how these variables interact is key to mastering capillary flow rate. This concept is widely used in fields such as medicine, biology, and engineering, where precise fluid control is needed.

Fluid Dynamics

Fluid dynamics is the study of how fluids behave when they are in motion. It encompasses a variety of phenomena and processes that occur when liquids move through different environments.

• Poiseuille’s Law is a critical equation in the area of fluid dynamics, especially when examining the flow in narrow tubes like capillaries.
• This field studies factors including velocity, pressure changes, flow type (laminar vs. turbulent), and viscosity.
• In narrow tubes, assuming laminar flow (which is orderly and smooth), fluid dynamics allows us to predict and control the flow rate depending on the parameters mentioned in Poiseuille's equation.

Mastering fluid dynamics helps in designing systems for efficient fluid transport, managing pipelines, and understanding natural processes such as blood flow in animals.

Pressure Difference in Liquids

The pressure difference, denoted as \( \Delta P \), is a driving force for liquid flow in tubes. It refers to the variance in pressure from the entry to the exit of the tube, pushing the fluid through it.

• The greater the pressure difference, the higher the potential flow rate of the fluid, given the other conditions remain constant.
• In the context of Poiseuille’s Law, \( \Delta P \) is a proportional factor that directly affects the flow rate. More pressure difference means more force available to drive the fluid along.
• Pressure differences occur naturally in many systems, or they can be introduced mechanically.

Controlling pressure differences is essential in processes ranging from industrial applications to biological systems, where precise movement of fluids determines the efficiency and functioning of the system.